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Everyone needs 2
sheets of oragami
paper on the counter
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5.6.1: Proving the Interior Angle Sum Theorem
What is the name of this triangle?
(there could be two names)
Answer:
Obtuse and/or
Scalene
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5.6.1: Proving the Interior Angle Sum Theorem
What is the name of this triangle?
Answer:
Right Triangle
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5.6.1: Proving the Interior Angle Sum Theorem
What is the name of this triangle?
Answer:
Acute Triangle
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5.6.1: Proving the Interior Angle Sum Theorem
What is the name of this triangle?
Answer:
Isosceles Triangle
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5.6.1: Proving the Interior Angle Sum Theorem
What is the name of this triangle?
Answer:
Equilateral Triangle
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5.6.1: Proving the Interior Angle Sum Theorem
Key Concepts, continued
• All of the angles of an acute triangle are acute,
or less than 90°.
• One angle of an obtuse triangle is obtuse, or
greater than 90°.
• A right triangle has one angle that
measures 90°.
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5.6.1: Proving the Interior Angle Sum Theorem
Key Concepts, continued
Acute triangle
Obtuse triangle
Right triangle
All angles
are less than 90°.
One angle
is greater than 90°.
One angle
measures 90°.
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5.6.1: Proving the Interior Angle Sum Theorem
Key Concepts, continued
• Triangles classified by the number of congruent sides
can be scalene, isosceles, or equilateral.
• A scalene triangle has no congruent sides.
• An isosceles triangle has at least two
congruent sides.
• An equilateral triangle has three congruent
sides.
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5.6.1: Proving the Interior Angle Sum Theorem
Key Concepts, continued
Scalene triangle
Isosceles triangle
Equilateral triangle
No congruent sides
At least two
congruent sides
Three congruent
sides
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5.6.1: Proving the Interior Angle Sum Theorem
What do these two figures have in
common?
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5.6.1: Proving the Interior Angle Sum Theorem
Key Concepts, continued
• It is possible to create many different triangles, but
the sum of the angle measures of
every triangle is 180°. This is known as the
Triangle Sum Theorem.
• PROVE IT!!
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5.6.1: Proving the Interior Angle Sum Theorem
Key Concepts, continued
Theorem
Triangle Sum Theorem
The sum of the angle measures of a triangle is 180°.
**m∠A + m∠B + m∠C = 180
5.6.1: Proving the Interior Angle Sum Theorem
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Key Concepts, continued
• Interior angles are the angles inside the triangle.
• Exterior angles are angles formed by one side of the
triangle and the extension of another side.
• The interior angles that are not adjacent to the
exterior angle are called the remote interior angles
of the exterior angle.
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5.6.1: Proving the Interior Angle Sum Theorem
Key Concepts, continued
• Interior angles: ∠A,
∠B, and ∠C
• Exterior angle: ∠D
• Remote interior
angles of ∠D: ∠A and
∠B
•
Notice that ∠C and ∠D are supplementary; that is,
together they create a line and sum to 180°.
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5.6.1: Proving the Interior Angle Sum Theorem
Key Concepts, continued
Theorem
Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the
sum of the measures of its remote interior angles.
**m∠D = m∠A + m∠B
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5.6.1: Proving the Interior Angle Sum Theorem
Key Concepts, continued
Theorem
If one angle of a triangle has a greater measure than another
angle, then the side opposite the greater angle is longer than
the side opposite the lesser angle.
m∠A < m∠B < m∠C
a<b<c
5.6.1: Proving the Interior Angle Sum Theorem
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Guided Practice
Example 1 pg 281
Find the measure of ∠C.
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5.6.1: Proving the Interior Angle Sum Theorem
Guided Practice: Example 1, continued
1. Identify the known information.
Two measures of the three interior angles are given
in the problem.
m∠A = 80
m∠B = 65
The measure of ∠C is unknown.
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5.6.1: Proving the Interior Angle Sum Theorem
Guided Practice: Example 1, continued
2. Calculate the measure of ∠C.
The sum of the measures of the interior angles of a
triangle is 180°.
Create an equation to solve for the unknown measure
of ∠C.
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5.6.1: Proving the Interior Angle Sum Theorem
Guided Practice: Example 1, continued
m∠A + m∠B + m∠C = 180
Triangle Sum Theorem
80 + 65 + m∠C = 180
Substitute values for m∠A
and m∠B.
145 + m∠C = 180
Simplify.
m∠C = 35
Solve for m∠C.
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5.6.1: Proving the Interior Angle Sum Theorem
Guided Practice: Example 1, continued
3. State the answer.
The measure of ∠C is 35°.
✔
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5.6.1: Proving the Interior Angle Sum Theorem
Guided Practice
Example 2 pg 282
Find the missing
angle measures.
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5.6.1: Proving the Interior Angle Sum Theorem
Guided Practice: Example 2, continued
m∠A + m∠B + m∠BCA = 180
Triangle Sum
Theorem
50 + 55 + m∠BCA = 180
Substitute values for
m∠A and m∠B.
105 + m∠BCA = 180
Simplify.
m∠BCA = 75
Solve for m∠BCA.
∠BCA and ∠DCE are vertical angles and are
congruent.
m∠DCE = m∠BCA = 75
5.6.1: Proving the Interior Angle Sum Theorem
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Guided Practice: Example 2, continued
Create an equation to solve for the unknown
measure of ∠D.
m∠DCE + m∠D + m∠E = 180
Triangle Sum
Theorem
75 + m∠D + 40 = 180
Substitute values for
m∠DCE and m∠E.
115 + m∠D = 180
Simplify.
m∠D = 65
Solve for m∠D.
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5.6.1: Proving the Interior Angle Sum Theorem
Guided Practice: Example 2, continued
3. State the answer.
The measure of ∠BCA is 75°.
The measure of ∠DCE is 75°.
The measure of ∠D is 65°.
✔
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5.6.1: Proving the Interior Angle Sum Theorem